Zero product preserving maps of operator-valued functions

Let be locally compact Hausdorff spaces and , be Banach algebras. Let be a zero product preserving bounded linear map with dense range. We show that is given by a continuous field of algebra homomorphisms from into if is irreducible. As corollaries, such a surjective arises from an algebra homomorphism, provided that is a -algebra and is a semi-simple Banach algebra, or both and are -algebras.

     

Control of radii of convergence and extension of subanalytic functions

Let : denote a real analytic function on an open subset of , and let denote the points where does not admit a local analytic extension. We show that if is semialgebraic (respectively, globally subanalytic), then is semialgebraic (respectively, subanalytic) and extends to a semialgebraic (respectively, subanalytic) neighbourhood of . (In the general subanalytic case, is not necessarily subanalytic.) Our proof depends on controlling the radii of convergence of power series centred at points in the image of an analytic mapping , in terms of the radii of convergence of at points , where denotes the Taylor expansion of at .

     

A classification of rapidly growing Ramsey functions

Let be a number-theoretic function. A finite set of natural numbers is called -large if . Let be the Paris Harrington statement where we replace the largeness condition by a corresponding -largeness condition. We classify those functions for which the statement is independent of first order (Peano) arithmetic . If is a fixed iteration of the binary length function, then is independent. On the other hand is provable in . More precisely let where denotes the -times iterated binary length of and denotes the inverse function of the -th member of the Hardy hierarchy. Then is independent of (for ) iff .

     

Mansfield's imprimitivity theorem for arbitrary closed subgroups

Let be a nondegenerate coaction of on a -algebra , and let be a closed subgroup of . The dual action is proper and saturated in the sense of Rieffel, and the generalised fixed-point algebra is the crossed product of by the homogeneous space . The resulting Morita equivalence is a version of Mansfield's imprimitivity theorem which requires neither amenability nor normality of .

     

Eigenvalue estimates for operators on symmetric Banach sequence spaces

Using abstract interpolation theory, we study eigenvalue distribution problems for operators on complex symmetric Banach sequence spaces. More precisely, extending two well-known results due to König on the asymptotic eigenvalue distribution of operators on -spaces, we prove an eigenvalue estimate for Riesz operators on -spaces with , which take values in a -concave symmetric Banach sequence space , as well as a dual version, and show that each operator on a -convex symmetric Banach sequence space , which takes values in a -concave symmetric Banach sequence space , is a Riesz operator with a sequence of eigenvalues that forms a multiplier from into . Examples are presented which among others show that the concavity and convexity assumptions are essential.

     

Exponential nonnegativity

Let be a Banach algebra, , the spectrum of and the spectral abscissa of . If , then we show that there exists an algebra cone such that is exponentially nonnegative with respect to and the spectral radius is increasing on .

     

Real rank and squaring mappings for unital -algebras

It is proved that if is a compact Hausdorff space of Lebesgue dimension , then the squaring mapping , defined by , is open if and only if . Hence the Lebesgue dimension of can be detected from openness of the squaring maps . In the case it is proved that the map , from the selfadjoint elements of a unital -algebra into its positive elements, is open if and only if is isomorphic to for some compact Hausdorff space with .

     

The free roots of the complete graph

There is a model-completion of the theory of a (reflexive) -coloured graph such that is total, and for all . For 2$">, the theory is not simple, and does not have the strict order property. The theories combine to yield a non-simple theory without the strict order property, which does not eliminate hyperimaginaries.

     

Critical exponents of discrete groups and -spectrum

Let be a noncompact semisimple Lie group and an arbitrary discrete, torsion-free subgroup of . Let be the bottom of the spectrum of the Laplace-Beltrami operator on the locally symmetric space , and let be the exponent of growth of . If has rank , then these quantities are related by a well-known formula due to Elstrodt, Patterson, Sullivan and Corlette. In this note we generalize that relation to the higher rank case by estimating from above and below by quadratic polynomials in . As an application we prove a rigiditiy property of lattices.

     

On the invariance of classes under composition

The necessary and sufficient condition for to be in the class for every of that class whose range is in the domain of is that be in .

     

Minimal polynomials of elements of order in -modular projective representations of alternating groups

Let be an algebraically closed field of characteristic 0$"> and let be a quasi-simple group with . We describe the minimal polynomials of elements of order in irreducible representations of over . If , we determine the minimal polynomials of elements of order in -modular irreducible representations of , , , , , and .

     

Robust transitivity and topological mixing for -flows

We prove that nontrivial homoclinic classes of -generic flows are topologically mixing. This implies that given , a nontrivial -robustly transitive set of a vector field , there is a -perturbation of such that the continuation of is a topologically mixing set for . In particular, robustly transitive flows become topologically mixing after -perturbations. These results generalize a theorem by Bowen on the basic sets of generic Axiom A flows. We also show that the set of flows whose nontrivial homoclinic classes are topologically mixing is not open and dense, in general.

     

Rings with finite Gorenstein injective dimension

In this paper we prove that for any associative ring , and for any left -module with finite projective dimension, the Gorenstein injective dimension equals the usual injective dimension . In particular, if is finite, then also is finite, and thus is Gorenstein (provided that is commutative and Noetherian).

     

A concrete description of -spaces as -spaces and its applications

We prove that for a compact Hausdorff space without isolated points, and are isometrically Riesz isomorphic spaces under a certain topology on . Moreover, is a closed subspace of . This provides concrete examples of compact Hausdorff spaces such that the Dedekind completion of is (= the set of all bounded real-valued functions on ) since the Dedekind completion of is ( and spaces as Banach lattices).

     

Free products in linear groups

Let be a commutative integral domain of characteristic , and let be a finite subgroup of , the projective general linear group of degree over . In this note, we show that if , then also contains the free product , where is the infinite cyclic group generated by the image of a suitable transvection.

     

Self-normalizing Sylow subgroups

Using the classification of finite simple groups we prove the following statement: Let 3$"> be a prime, a group of automorphisms of -power order of a finite group , and a -invariant Sylow -subgroup of . If is trivial, then is solvable. An equivalent formulation is that if has a self-normalizing Sylow -subgroup with 3$"> a prime, then is solvable. We also investigate the possibilities when .

     

Strong comparison principle for solutions of quasilinear equations

Let , , be a bounded smooth connected open set and be a map satisfying the hypotheses (H1)-(H4) below. Let with , in and with be two weak solutions of

Suppose that in . Then we show that u_1$"> in under the following assumptions: either u_1$"> on , or on and in . We also show a measure-theoretic version of the Strong Comparison Principle.

     

Backward uniqueness for solutions of linear parabolic equations

We address the backward uniqueness property for the equation in . We show that under rather general conditions on and , implies that vanishes to infinite order for all points . It follows that the backward uniqueness holds if and when n/2$">. The borderline case is also covered with an additional continuity and smallness assumption.

     

On linear transformations preserving at least one eigenvalue

Let be an algebraically closed field and be a linear transformation. In this paper we show that if preserves at least one eigenvalue of each matrix, then preserves all eigenvalues of each matrix. Moreover, for any infinite field (not necessarily algebraically closed) we prove that if is a linear transformation and for any with at least an eigenvalue in , and have at least one common eigenvalue in , then preserves the characteristic polynomial.

     

The Cauchy problem for a class of Kovalevskian pseudo-differential operators

We prove the well-posedness of the forward Cauchy problem for a pseudo-differential operator of order with the Log-Lipschitz continuous symbol in the time variable. The characteristic roots of are distinct and satisfy the necessary Lax-Mizohata condition Im . The Log-Lipschitz regularity has been tested as the optimal one for well-posedness in the case of second-order hyperbolic operators. Our main aim is to present a simple proof which needs only a little of the basic calculus of standard pseudo-differential operators.